EXAMPLE 3 Calculating

Use the following steps to calculate the test statistic for the data in Table 2:

  1. Find , the slope of the regression line.
  2. Calculate , the standard error of the estimate.
  3. Compute , the numerator of the sample variance of the data.

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Solution

All calculations up to the final result are expressed to nine decimal places.

  1. From Example 1, the slope of the regression line is .

  2. Recall from Section 4.3 (page 228) that

    is the standard error of the estimate. Squaring each residual from Table 2 gives us the squared residuals in Table 3, and the sum of squared residuals, or sum of squares error, equal to

    Then the standard error of the estimate is

    Table 13.4: Table 3 Calculating SSE
    Residuals Squared residuals
    1.4 1.96
    −1.6 2.56
    −0.6 0.36
    3.4 11.56
    −2.6 6.76
    −2.6 6.76
    3.4 11.56
    −0.6 0.36
    −1.6 2.56
    1.4 1.96

  3. To compute , we note from page 110 in Chapter 3 that the sample variance of is

    Multiplying each side of the equation by , we obtain an equation for the quantity :

    The TI-83/84 output from Figure 7 shows that , and, because ,

    Therefore,

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    Figure 13.7: FIGURE 7 Summary statistics for the (age) data.